Ronald E. Parr wrote:
> When you apply Bayes rule, you are assuming that future outcomes will be
> drawn from the same distribution as past outcomes, i.e. you are assuming
> that induction works. Therefore, you cannot use Bayes rule to justify
> induction without falling into the trap of circular reasoning.
I have to disagree.
First of all, you are assuming a frequentist interpretation of probabilities
when you talk about drawing outcomes from a distribution. But the whole point
of the Bayesian approach is that the laws of probability theory apply equally
well to degrees of confidence/plausibility/belief -- you don't have to tie them
to limiting frequencies. Used in this way, a probability distribution encodes
one's state of information, and says nothing (directly) about frequencies.
Secondly, when you say "future outcomes will be drawn from the same
distribution...," are you implying some physical process here? This sounds
suspiciously like the old Mind Projection Fallacy again, wherein one assumes
that a mathematical construct (probability distributions) that describes the
internal state of information of one's mind is an actual physical property of
the external world.
Finally, Bayes' Rule does not depend on any assumption of independent variables
assigned identical distributions, as you seem to imply. Bayes' Rule is a
general rule that relates any pair of propositions.
